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A group having an infinite number of elements. Some infinite groups, such as the integers or rationals, are not continuous groups.

The sequence generated by repeated application of a map. The map is said to have a closed orbit if it has a finite number of elements.

In the plane, the reflection property can be stated as three theorems (Ogilvy 1990, pp. 73-77): 1. The locus of the center of a variable circle, tangent to a fixed circle and ...

A finite division algebra is a field.

Roughly speaking, a matroid is a finite set together with a generalization of a concept from linear algebra that satisfies a natural set of properties for that concept. For ...

Abstract Algebra

Let J be a finite group and the image R(J) be a representation which is a homomorphism of J into a permutation group S(X), where S(X) is the group of all permutations of a ...

The crystallographic point groups are the point groups in which translational periodicity is required (the so-called crystallography restriction). There are 32 such groups, ...

C_4 is one of the two groups of group order 4. Like C_2×C_2, it is Abelian, but unlike C_2×C_2, it is a cyclic. Examples include the point groups C_4 (note that the same ...

The dihedral group D_n is the symmetry group of an n-sided regular polygon for n>1. The group order of D_n is 2n. Dihedral groups D_n are non-Abelian permutation groups for ...